Subgroups — #70518

Order 384.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×#822	C₂	1	4	yes	maximal
3	#4924	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#1848	—	3	1	no	maximal, Sylow
6	C₂×C₃×M₁₆	C₂²	1	2	yes	—
7	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₂×#208	C₂²	1	2	yes	—
9	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
10	C₃×C₄⋊C₈	—	2	2	no	—
11	#822	—	2	8	no	—
12	C₄⋊(C₃⋊C₈)	—	2	2	no	—
13	C₄×C₃⋊C₈	C₄	1	2	yes	—
14	C₄×C₃⋊C₈	C₄	1	2	yes	—
15	C₂×#108	—	3	4	no	—
16	#423	—	3	1	no	—
17	#423	D₃	1	1	yes	characteristic
18	C₂×C₄×C₈	—	3	1	no	—
19	C₂²×C₃×C₄	Q₈	1	2	yes	—
20	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
21	C₃×M₁₆	—	2	4	no	—
22	C₂×C₃×C₈	—	2	2	no	—
23	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₃×C₄²	D₄	1	2	yes	—
26	#208	—	2	4	no	—
27	C₂×C₃⋊C₈	—	2	2	no	—
28	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
29	C₂×C₃⋊C₈	D₄	1	4	yes	—
30	C₂×C₃⋊C₈	Q₈	1	4	yes	—
31	C₂×M₁₆	—	3	2	no	—
32	C₂×M₁₆	C₂×D₃	1	2	yes	—
33	C₂²×C₈	—	3	2	no	—
34	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
35	#108	—	6	8	no	—
36	C₄⋊C₈	—	2	2	no	—
37	C₄⋊C₈	—	6	2	no	—
38	C₄×C₈	—	3	2	no	—
39	C₄×C₈	—	3	2	no	—
40	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₃×C₄	—	2	4	no	—
42	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
43	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
44	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
46	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
48	C₃×C₈	—	4	2	no	—
49	C₃⋊C₈	—	4	2	no	—
50	C₃⋊C₈	C₄⋊C₄	1	8	yes	—
51	C₂²×C₄	Dic₆	1	2	yes	—
52	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
53	M₁₆	—	2	4	no	—
54	M₁₆	—	6	4	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	3	4	no	—
57	C₂×C₈	—	3	4	no	—
58	C₂×C₈	—	3	4	no	—
59	C₂×C₈	—	6	2	no	—
60	C₄²	C₄×D₃	1	1	yes	characteristic
61	C₄²	C₄×D₃	1	1	yes	characteristic
62	C₄²	#67	1	2	yes	—
63	C₂²×C₃	—	2	2	no	—
64	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₃	C₄⋊Q₈	1	2	yes	—
66	C₃×C₄	—	2	2	no	—
67	C₃×C₄	#108	1	4	yes	complemented
68	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
69	C₃×C₄	#127	1	1	yes	characteristic
70	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
71	C₂³	C₂×Dic₆	1	1	yes	characteristic
72	C₂×C₄	—	2	4	no	—
73	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
74	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
75	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
76	C₂×C₄	D₃×D₄	1	2	yes	—
77	C₂×C₄	D₃×Q₈	1	2	yes	—
78	C₂×C₄	C₂×#67	1	1	yes	characteristic
79	C₈	—	3	8	no	—
80	C₈	—	4	2	no	—
81	C₈	—	12	2	no	—
82	C₂×C₃	—	2	2	no	—
83	C₂×C₃	#384	1	1	yes	characteristic
84	C₂×C₃	C₂×#108	1	2	yes	—
85	C₂²	—	2	2	no	—
86	C₂²	#888	1	2	yes	—
87	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	#822	1	4	yes	complemented
90	C₄	D₃×C₄⋊C₄	1	2	yes	—
91	C₄	#940	1	1	yes	characteristic
92	C₄	#944	1	1	yes	characteristic
93	C₃	#1848	1	1	yes	characteristic, complemented, Sylow
94	C₂	—	2	2	no	—
95	C₂	#4975	1	1	yes	characteristic
96	C₂	C₂×#822	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#423	C₂
socle	C₂²×C₃	C₂×C₄⋊C₄
prime radical	C₂³×C₃	C₂×Q₈
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#423	D₃
hypocenter	C₃	#1848
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
67. C₃×C₄	#108	semidirect
89. C₄	#822	semidirect
93. C₃	#1848	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#423	2
C₂×#822	2
#4924	2
C₂×C₄×C₃⋊C₈	2
#1848	3